With a unique kinematic arrangement, a new type of quadruped robot with reduced degrees of freedom (DoF) requires minimal-torque actuators to achieve high-payload locomotion. This paper focuses on the kinematic analysis and design optimization for robots of this type. To plan and control its change of posture, a necessary strategy to find feasible solutions of full-body inverse kinematics under additional kinematic constraints is introduced. A design method via nonlinear programming (NLP) is first presented in order to optimize link parameters with guarantee to a series of successive steps. Workspace is also investigated to prepare for further dynamic motion planning. We have verified feasibility of proposed methods with software simulations and hardware implementations, e.g., omni-directional walking and situ rotation.